By employing a combined method of density functional theory and dynamical mean field theory $(\mathrm{DFT}+\mathrm{DMFT})$ we investigate the effect of electronic correlations on the magnetic and superconducting properties of the iron-based parent compound LaFeAsO. We find that the static nonlocal susceptibility $\chi \left(\mathbf{q}\right)$ and the dynamical spin structure factor $S(\mathbf{q},\omega )$ exhibit a peak at the in-plane wave vector $\mathbf{Q}=(\pi ,\pi )$, which is strongly enhanced upon inclusion of dynamical vertex corrections in the ladder approximation, leading to magnetic instability. Considering the eigenfunctions of the Bethe-Salpeter equation with the vertex, obtained within the second-order perturbation theory, as well as the ladder approach containing dynamic interaction vertices, we obtain a close competition between $d$-wave and $s_{\pm }$ order parameters, dominating in the second-order and ladder approach, respectively. We argue that the dominating $s_{\pm }$ instability in the ladder $\mathrm{DFT}+\mathrm{DMFT}$ approach is related to the reduced degree of magnetic frustration by itinerant degrees of freedom due to only partially formed local magnetic moments. Our study shows that dynamic correlation effects do not change the type of leading superconducting instability in LaFeAsO.

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• Density of states
• Electronic structure
• Fermi surface
• Superconducting gap
• Superconducting order parameter
• Iron-based superconductors
• Strongly correlated systems
• Density functional theory
• Dynamical mean field theory
• Electron-correlation calculations
• Quantum Monte Carlo